1895.] M7' J. Larmor, On Graphical Methods. 307 



Monday, 27 May, 1895. 



Professor J. J. Thomson President, in the Chair. 



The following Communications were made to the Society : 



(1) On Graphical Methods in Geometrical Optics. By J. 

 Larmor, M.A., St John's College. 



1. The fundamental problem of geometrical optics relates to 

 the modification produced in a filament of light on passing across 

 a series of different media. From the geometrical standpoint 

 the filament is made up of a narrow pencil of rays, which are 

 straight when the medium is homogeneous : and it may be con- 

 sidered as defined by the focal lines of the pencil. The direct 

 analytical method therefore hinges on the determination of these 

 focal lines, after each refraction, from their already known positions 

 before refraction : but the formulae, even for a single refraction, 

 are complicated, and, when there is a question of combining a 

 number of successive refractions, almost prohibitive. In such 

 circumstances however, as in other physical questions where we 

 have to do with linear relations, the use of graphic methods will 

 be found to lead to results of intrinsic simplicity, and thus give 

 direct insight into the general relations of the subject. 



2. Uniplanar System. In the sketch here to be given of a 

 geometrical method of treatment, it will be convenient to begin 

 with the simple case of a narrow pencil of rays in one plane, and 

 having therefore only one focus: that is, the optical system will at 

 first be a columnar or cylindrical one. 



Since a single focus now always corresponds to a single focus as 

 image to object, and, the pencil being narrow, all its rays may 

 be taken to a sufficient approximation as passing exactly through 

 the focus, the elementary geometrical theory of Mobius and 

 Maxwell may be applied to the filament. Thus we may determine 

 on its axes at incidence and emergence a pair of principal points, 

 and a pair of principal foci, and we may construct by aid of 

 them the focus conjugate to any other assigned one \ 



In this way, or by the more usual analytical processes, we see 

 that conjugate foci are connected by a linear relation, as might 

 have been expected a priori. Now let us draw the axis of 

 the filament as incident on the optical system, and its axis as 

 emergent from it. Conjugate foci on these two axes are homo- 

 graphically related ; therefore the line connecting any pair of them 

 envelopes a conic section which also has contact with the axes 

 themselves. Suppose, by experiment or otherwise, that three pairs 

 of conjugate foci have been determined: it will be possible to 



1 Cf. Proc. Lond. Math. Soc, xx., 1889, p. 182. 



23—2 



